Introduction to Chemical Kinetics and Reaction Mechanisms

Introduction to Chemical Kinetics

• Kinetics Defined: The study of the rate (speed) at which a chemical process occur and the reaction mechanism (the exact sequence of steps).

• Measuring Rates: Determined by monitoring the change in concentration of reactants or products as a function of time.

• Types of Rates:

  • Average Rate: The change in concentration over a specific time interval ($\text{Average rate} = \frac{\Delta[C_4H_9Cl]}{\Delta t}$).

  • Instantaneous Rate: The slope of the line tangent to the concentration-time curve at any specific point; best measured near the start of a reaction.

Factors Affecting Reaction Rates

• Physical State: Molecules must come into contact to react. Homogeneous mixtures allow for faster reactions than heterogeneous ones.

• Concentration of Reactants: Higher concentrations increase the frequency of molecular collisions.

• Temperature: Increased temperature provides reactant molecules with more kinetic energy, leading to more frequent and higher-energy collisions.

• Presence of a Catalyst: Catalysts speed up reactions by changing the reaction mechanism and are not consumed in the process.

Reaction Rates and Stoichiometry

• 1:1 Ratios: For a reaction like $C_4H_9Cl(aq) + H_2O(l) \rightarrow C_4H_9OH(aq) + HCl(aq)$, the rate of disappearance of reactant equals the rate of appearance of product:

  • $\text{Rate} = -\frac{\Delta[C_4H_9Cl]}{\Delta t} = \frac{\Delta[C_4H_9OH]}{\Delta t}$

• General Stoichiometry: For the general reaction $aA + bB \rightarrow cC + dD$:

  • $\text{Rate} = -\frac{1}{a} \frac{\Delta[A]}{\Delta t} = -\frac{1}{b} \frac{\Delta[B]}{\Delta t} = \frac{1}{c} \frac{\Delta[C]}{\Delta t} = \frac{1}{d} \frac{\Delta[D]}{\Delta t}$

Rate Laws and Concentration

• Rate Law Expression: Shows the relationship between rate and reactant concentrations: $\text{Rate} = k[A]^m[B]^n$.

• Rate Constant ($k$): A temperature-dependent constant unique to a reaction.

• Reaction Orders: The exponents ($m$ and $n$) indicate the order with respect to each reactant. The sum of exponents is the overall reaction order.

• Example Case: For $NH_4^+(aq) + NO_2^-(aq) \rightarrow N_2(g) + 2H_2O(l)$, the rate law is $\text{Rate} = k[NH_4^+][NO_2^-]$ (second-order overall).

Integrated Rate Laws and Half-Life

• First-Order Rate Law:

  • Equation: $\ln[A]_t = -kt + \ln[A]_0$

  • Linear Plot: A plot of $\ln[A]$ vs. $t$ yields a straight line with slope $-k$.

  • Half-Life ($t_{1/2}$): The time for half the reactant to be consumed. For first-order, $t_{1/2} = \frac{0.693}{k}$ (independent of initial concentration).

• Second-Order Rate Law:

  • Equation: $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$

  • Linear Plot: A plot of $\frac{1}{[A]}$ vs. $t$ yields a straight line with slope $k$.

  • Half-Life ($t_{1/2}$): Depends on initial concentration: $t_{1/2} = \frac{1}{k[A]_0}$.

Activation Energy and Temperature

• Collision Model: Reactants must collide with correct orientation and sufficient energy to break/form bonds.

• Activation Energy ($E_a$): The minimum energy required for a reaction to occur.

• Transition State: The high-energy species (activated complex) present at the activation-energy barrier.

• Maxwell-Boltzmann Distribution: At higher temperatures, a larger fraction ($f = e^{-E_a/RT}$) of molecules possesses energy exceeding $E_a$.

• Arrhenius Equation: Relates the rate constant to temperature and activation energy:

  • $k = Ae^{-E_a/RT}$

  • Linear Form: $\ln(k) = -\frac{E_a}{R} \left( \frac{1}{T} \right) + \ln(A)$

Reaction Mechanisms and Catalysis

• Elementary Reactions: Single discrete steps in a mechanism.

• Molecularity: Number of molecules involved in an elementary step (unimolecular, bimolecular, termolecular).

• Rate-Determining Step: The slowest step in a multistep mechanism that limits the overall reaction rate.

• Intermediates: Species produced in one step and consumed in a later step (e.g., $NO_3$, $NOBr_2$).

• Catalysts: Increase rate by lowering $E_a$ and changing the mechanism.

• Enzymes: Biological catalysts where a substrate fits into an active site (lock-and-key model).

Questions & Discussion

• Q: How is the rate of ozone disappearance related to oxygen appearance in $2O_3(g) \rightarrow 3O_2(g)$?

• A: $\text{Rate} = -\frac{1}{2} \frac{\Delta[O_3]}{\Delta t} = \frac{1}{3} \frac{\Delta[O_2]}{\Delta t}$. If $\frac{\Delta[O_2]}{\Delta t} = 6.0 \times 10^{-5}\,M/s$, then $-\frac{\Delta[O_3]}{\Delta t} = \frac{2}{3}(6.0 \times 10^{-5}\,M/s) = 4.0 \times 10^{-5}\,M/s$.

• Q: What is the concentration of a first-order insecticide ($k = 1.45\,yr^{-1}$) after 1 year if initially $5.0 \times 10^{-7}\,g/cm^3$?

• A: Using $\ln[A]_t = -(1.45)(1.0) + \ln(5.0 \times 10^{-7})$, the concentration remaining is $1.2 \times 10^{-7}\,g/cm^3$.

• Q: Determine the rate law and $k$ for $2NO(g) + 2H_2(g) \rightarrow N_2(g) + 2H_2O(g)$ given experiment data.

• A: Rate law is $\text{Rate} = k[NO]^2[H_2]$ with $k = 1.2\,M^{-2}\,s^{-1}$.

• Q: Rank reaction mixtures in order of increasing rate for $\text{Rate} = k[A][B]^2$ (Box 1: 5 red/5 purple; Box 2: 7 red/3 purple; Box 3: 3 red/7 purple).

• A: Box 2 ($63k$) < Box 1 ($125k$) < Box 3 ($147k$). Concentration of B has more influence because it is second-order.